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CHAPTER 14 SECTION 1: ANALYSIS OF VARIANCE MULTIPLE CHOICE 1. In one-way ANOVA, the amount of total variation that is unexplained is measured by the: a. sum of squares for treatments. b. sum of squares for error. c. total sum of squares. d. degrees of freedom. ANS: B PTS: 1 REF: SECTION 14.1 2. The test statistic of the single-factor ANOVA equals: a. sum of squares for treatments / sum of squares for error. b. sum of squares for error / sum of squares for treatments. c. mean square for treatments / mean square for error. d. mean square for error / mean square for treatments. ANS: C PTS: 1 REF: SECTION 14.1 3. In a single-factor analysis of variance, MST is the mean square for treatments and MSE is the mean square for error. The null hypothesis of equal population means is rejected if: a. MST is much smaller than MSE. b. MST is much larger than MSE. c. MST is equal to MSE. d. None of these choices. ANS: B PTS: 1 REF: SECTION 14.1 4. Which of the following is not a required condition for one-way ANOVA? a. The sample sizes must be equal. b. The populations must all be normally distributed. c. The population variances must be equal. d. The samples for each treatment must be selected randomly and independently. ANS: A PTS: 1 REF: SECTION 14.1 5. The analysis of variance is a procedure that allows statisticians to compare two or more population: a. means. b. proportions. c. variances. d. standard deviations. ANS: A PTS: 1 REF: SECTION 14.1 6. The distribution of the test statistic for analysis of variance is the: a. normal distribution. b. Student t-distribution. c. F-distribution. d. None of these choices. ANS: C PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 7. In a one-way ANOVA, error variability is computed as the sum of the squared errors, SSE, for all values of the response variable. This variability is the: a. the total variation. b. within-treatments variation. c. between-treatments variation. d. None of these choices. ANS: B PTS: 1 REF: SECTION 14.1 8. In the one-way ANOVA where there are k treatments and n observations, the degrees of freedom for the F-statistic are equal to, respectively: a. n and k. b. k and n. c. n k and k 1. d. k 1 and n k. ANS: D PTS: 1 REF: SECTION 14.1 9. In the one-way ANOVA where k is the number of treatments and n is the number of observations in all samples, the degrees of freedom for treatments is given by: a. k 1 b. n k c. n 1 d. n k + 1 ANS: A PTS: 1 REF: SECTION 14.1 10. In ANOVA, the F-test is the ratio of two sample variances. In the one-way ANOVA (completely randomized design), the variance used as a numerator of the ratio is: a. mean square for treatments. b. mean square for error. c. total sum of squares. d. None of these choices. ANS: A PTS: 1 REF: SECTION 14.1 11. In a completely randomized design for ANOVA, the numerator and denominator degrees of freedom are 4 and 25, respectively. The total number of observations must equal: a. 24 b. 25 c. 29 d. 30 ANS: D PTS: 1 REF: SECTION 14.1 12. The number of degrees of freedom for the denominator in one-way ANOVA test involving 4 population means with 15 observations sampled from each population is: a. 60 b. 19 c. 56 d. 45 ANS: C PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 13. The value of the test statistic in a completely randomized design for ANOVA is F = 6.29. The degrees of freedom for the numerator and denominator are 5 and 10, respectively. Using an F table, the most accurate statements to be made about the p-value is that it is: a. greater than 0.05 b. between 0.025 and 0.050. c. between 0.010 and 0.025. d. between 0.001 and 0.010. ANS: D PTS: 1 REF: SECTION 14.1 14. In one-way ANOVA, the term refers to the: a. sum of the sample means. b. sum of the sample means divided by the total number of observations. c. sum of the population means. d. weighted average of the sample means. ANS: D PTS: 1 REF: SECTION 14.1 15. For which of the following is not a required condition for ANOVA? a. The populations are normally distributed. b. The population variances are equal. c. The samples are independent. d. All of these choices are required conditions for ANOVA. ANS: D PTS: 1 REF: SECTION 14.1 16. One-way ANOVA is applied to independent samples taken from three normally distributed populations with equal variances. Which of the following is the null hypothesis for this procedure? a. 1 + 2 + 3 = 0 b. 1 + 2 + 3 0 c. 1 = 2 = 3 = 0 d. 1 = 2 = 3 ANS: D PTS: 1 REF: SECTION 14.1 17. In the one-way ANOVA where k is the number of treatments and n is the number of observations in all samples, the number of degrees of freedom for error is: a. k 1 b. n k c. n 1 d. n k + 1 ANS: B PTS: 1 REF: SECTION 14.1 18. How does conducting multiple t-tests compare to conducting a single F-test? a. Multiple t-tests increases the chance of a Type I error. b. Multiple t-tests decreases the chance of a Type I error. c. Multiple t-tests does not affect the chance of a Type I error. d. This comparison cannot be made without knowing the number of populations. ANS: A PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 19. In one-way analysis of variance, between-treatments variation is measured by the: a. SSE b. SST c. SS(Total) d. standard deviation ANS: B PTS: 1 REF: SECTION 14.1 20. One-way ANOVA is applied to independent samples taken from four normally distributed populations with equal variances. If the null hypothesis is rejected, then we can infer that a. all population means are equal. b. all population means differ. c. at least two population means are equal. d. at least two population means differ. ANS: D PTS: 1 REF: SECTION 14.1 21. Consider the following partial ANOVA table: Source of Variation Treatments Error Total SS 75 60 135 df * * 19 MS 25 3.75 F 6.67 The numerator and denominator degrees of freedom for the F-test (identified by asterisks) are a. 4 and 15 b. 3 and 16 c. 15 and 4 d. 16 and 3 ANS: B PTS: 1 REF: SECTION 14.1 22. Consider the following ANOVA table: Source of Variation Treatments Error Total SS 4 30 34 df 2 12 14 MS 2.0 2.5 F 0.80 The number of treatments is a. 13 b. 5 c. 3 d. 12 ANS: C PTS: 1 REF: SECTION 14.1 23. In one-way analysis of variance, within-treatments variation is measured by: a. sum of squares for error. b. sum of squares for treatments. c. total sum of squares. d. standard deviation. ANS: A PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 24. Consider the following ANOVA table: Source of Variation Treatments Error Total SS 128 270 398 df 4 25 29 MS 32 10.8 F 2.963 The total number of observations is: a. 25 b. 29 c. 30 d. 32 ANS: C PTS: 1 REF: SECTION 14.1 25. In one-way analysis of variance, if all the sample means are equal, then the: a. total sum of squares is zero. b. sum of squares for error is zero. c. sum of squares for treatments is zero. d. sum of squares for error equals sum of squares for treatments. ANS: C PTS: 1 REF: SECTION 14.1 26. Which of the following components in an ANOVA table is not additive? a. Sum of squares b. Degrees of freedom c. Mean squares d. All of these choices are additive. ANS: C PTS: 1 REF: SECTION 14.1 27. In which case can an F-test be used to compare two population means? a. For one tail tests only. b. For two tail tests only. c. For either one or two tail tests. d. None of these choices. ANS: B PTS: 1 REF: SECTION 14.1 28. The F-test statistic in a one-way ANOVA is equal to: a. MST/MSE b. SST/SSE c. MSE/MST d. SSE/SST ANS: A PTS: 1 REF: SECTION 14.1 29. The numerator and denominator degrees of freedom for the F-test in a one-way ANOVA are, respectively, a. (n k) and (k 1) b. (k 1) and (n k) c. (k n) and (n 1) d. (n 1) and (k n) ANS: B PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 30. Which of the following statements is false? a. F = t2 b. The F-test can be used instead of a two tail t-test when you compare two population means. c. Doing three t-tests is statistically equivalent to doing one F-test when you compare three population means. d. All of these choices are true. ANS: C PTS: 1 REF: SECTION 14.1 TRUE/FALSE 31. We use the analysis of variance (ANOVA) technique to compare two or more population means. ANS: T PTS: 1 REF: SECTION 14.1 32. The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample means are equal. ANS: T PTS: 1 REF: SECTION 14.1 33. The analysis of variance (ANOVA) technique analyzes the variance of the data to determine whether differences exist between the population means. ANS: T PTS: 1 REF: SECTION 14.1 34. Conducting t-tests for each pair or population means is statistically equivalent to conducting one F-test comparing all the population means. ANS: T PTS: 1 REF: SECTION 14.1 35. The sum of squares for error is also known as the between-treatments variation. ANS: F PTS: 1 REF: SECTION 14.1 36. The F-test used in one-way ANOVA is an extension of the t-test of 1 2. ANS: T PTS: 1 REF: SECTION 14.1 37. In one-way ANOVA, the total variation SS(Total) is partitioned into two sources of variation: the sum of squares for treatments (SST) and the sum of squares for error (SSE). ANS: T PTS: 1 REF: SECTION 14.1 38. In ANOVA, a criterion by which the populations are classified is called a factor. ANS: T PTS: 1 REF: SECTION 14.1 39. We can use the F-test to determine whether 1 is greater than 2. ANS: F PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 40. In one-way ANOVA, the test statistic is defined as the ratio of the mean square for error (MSE) and the mean square for treatments (MST), namely, F = MSE / MST. ANS: F PTS: 1 REF: SECTION 14.1 41. The sum of squares for treatments (SST) is the variation attributed to the differences between the treatment means, while the sum of squares for error (SSE) measures the within-treatment variation. ANS: T PTS: 1 REF: SECTION 14.1 42. The F-test in ANOVA tests whether or not the population variances are equal. ANS: F PTS: 1 REF: SECTION 14.1 43. If the numerator (MST) degrees of freedom is 3 and the denominator (MSE) degrees of freedom is 18, the total number of observations must equal 21. ANS: F PTS: 1 REF: SECTION 14.1 44. The F-statistic in a one-way ANOVA represents the variation within the treatments divided by the variation between the treatments. ANS: F PTS: 1 REF: SECTION 14.1 45. The sum of squares for error (SSE) measures the amount of variation that is explained by the ANOVA model, while the sum of squares for treatments (SST) measures the amount of variation that remains unexplained. ANS: F PTS: 1 REF: SECTION 14.1 46. The distribution of the test statistic for analysis of variance is the F-distribution. ANS: T PTS: 1 REF: SECTION 14.1 47. The analysis of variance (ANOVA) tests hypotheses about population variances and requires all the population means to be equal. ANS: F PTS: 1 REF: SECTION 14.1 48. The F-test in ANOVA is an expansion of the t-test for two independent population means. ANS: T PTS: 1 REF: SECTION 14.1 49. When the F-test is used for ANOVA, the rejection region is always in the right tail. ANS: T PTS: 1 REF: SECTION 14.1 50. The within-treatments variation provides a measure of the amount of variation in the response variables that is caused by the treatments. ANS: F PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. COMPLETION 51. The ANOVA procedure tests to determine whether differences exist between two or more population ____________________. ANS: means PTS: 1 REF: SECTION 14.1 52. The null hypothesis of ANOVA is that all the population means are ____________________. ANS: equal PTS: 1 REF: SECTION 14.1 53. The alternative hypothesis of ANOVA is that ____________________ population means are different. ANS: at least 2 at least two two or more 2 or more PTS: 1 REF: SECTION 14.1 54. In ANOVA the populations are classified according to one or more criterion, called ____________________. ANS: factors PTS: 1 REF: SECTION 14.1 55. SST measures the variation ____________________ treatments. ANS: between PTS: 1 REF: SECTION 14.1 56. SSE measures the variation ____________________ treatments. ANS: within PTS: 1 REF: SECTION 14.1 57. The F-test statistic in ANOVA is equal to MS____________________ divided by MS____________________ and H0 is rejected for ____________________ values of F. ANS: T; E; large PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 58. If SST explains a significant portion of the total variation, we conclude that the population means ____________________ (do/do not) differ. ANS: do PTS: 1 REF: SECTION 14.1 59. The F-test in ANOVA requires that the random variable be ____________________ distributed with equal ____________________. ANS: normally; variances normal; variances PTS: 1 REF: SECTION 14.1 60. If we square the t-statistic for two means, the result is the ____________________-statistic. ANS: F PTS: 1 REF: SECTION 14.1 SHORT ANSWER TV Viewing Habits A statistician employed by a television rating service wanted to determine if there were differences in television viewing habits among three different cities in California. She took a random sample of five adults in each of the cities and asked each to report the number of hours spent watching television in the previous week. The results are shown below. (Assume normal distributions with equal variances.) Hours Spent Watching Television San Diego Los Angeles San Francisco 25 28 23 31 33 18 18 35 21 23 29 17 27 36 15 61. {TV Viewing Habits Narrative} Set up the ANOVA Table. Use = 0.05 to determine the critical value. ANS: Source of Variation Treatments Error Total PTS: 1 SS 450.533 184.400 634.933 df 2 12 14 MS 225.267 15.367 F 14.659 P-value 0.0006 F critical 3.885 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 62. {TV Viewing Habits Narrative} Can she infer at the 5% significance level that differences in hours of television watching exist among the three cities? ANS: H0: 1 = 2 = 3 vs. H1: At least two means differ Conclusion: Reject the null hypothesis. Yes, differences in mean hours of television watching exist in at least two of the three cities, according to this data. PTS: 1 REF: SECTION 14.1 Pain Formulas A pharmaceutical manufacturer has been researching new formulas to provide quicker relief of minor pains. Their laboratories have produced three different formulas and they want to determine if the different formulas produce different responses. Fifteen people who complained of minor pains were recruited for an experiment; five were randomly assigned to each formula. Each person was asked to take the medicine and report the length of time until some relief was felt (minutes). The results are shown below. (Assume normal distributions with equal variances.) Time in Minutes Until Relief Is Felt (min) Formula 1 Formula 2 Formula 3 4 2 6 8 5 7 6 3 7 9 7 8 8 1 6 63. {Pain Formulas Narrative} Set up the ANOVA Table. Use = 0.05 to determine the critical value. ANS: Source of Variation Treatments Error Total PTS: 1 SS 36.4 42.0 78.4 df 2 12 14 MS 18.2 3.5 F 5.2 P-value 0.0236 F critical 3.885 REF: SECTION 14.1 64. {Pain Formulas Narrative} Do these data provide sufficient evidence to indicate that differences in the average time of relief exist among the three formulas? Use = 0.05. ANS: H0: 1 = 2 = 3 vs. H1: At least two means differ Conclusion: Reject the null hypothesis. Yes, differences in the mean time of relief exist in at least two of the three formulas, according to this data. PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. Pizza Customers The marketing manager of a pizza chain is in the process of examining some of the demographic characteristics of her customers. In particular, she would like to investigate the belief that the ages of the customers of pizza parlors, hamburger emporiums, and fast-food chicken restaurants are different. The ages of eight randomly selected customers of each of the restaurants are recorded and listed below. From previous analyses we know that the ages are normally distributed with equal variances for each group. Pizza 23 19 25 17 36 25 28 31 Customers' Ages Hamburger 26 20 18 35 33 25 19 17 Chicken 25 28 36 23 39 27 38 31 65. {Pizza Customers Narrative} Set up the ANOVA Table. Use = 0.05 to determine the critical value. ANS: Source of Variation Treatments Error Total PTS: 1 SS 203.583 863.750 1067.333 df 2 21 23 MS 101.792 41.131 F 2.475 P-value 0.1084 F critical 3.467 REF: SECTION 14.1 66. {Pizza Customers Narrative} Do these data provide enough evidence at the 5% significance level to infer that there are differences in ages among the customers of the three restaurants? ANS: H0: 1 = 2 = 3 vs. H1: At least two means differ Conclusion: Don't reject the null hypothesis. Cannot conclude that average age differences exist among the customers of the three restaurants, according to this data. PTS: 1 REF: SECTION 14.1 GMAT Scores A recent college graduate is in the process of deciding which one of three graduate schools he should apply to. He decides to judge the quality of the schools on the basis of the Graduate Management Admission Test (GMAT) scores of those who are accepted into the school. A random sample of six students in each school produced the following GMAT scores. Assume that the data are normally distributed with equal variances for each school. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. School 1 650 620 630 580 710 690 GMAT Scores School 2 105 550 700 630 600 650 School 3 590 510 520 500 490 530 67. {GMAT Scores Narrative} Set up the ANOVA Table. Use = 0.05 to determine the critical value. ANS: Source of Variation Treatments Error Total PTS: 1 SS 47,511 41,400 88,911 df 2 15 17 MS 23,756 2,760 F 8.61 P-value 0.003 F critical 2.70 REF: SECTION 14.1 68. {GMAT Scores Narrative} Can he infer at the 10% significance level that the GMAT scores differ among the three schools? ANS: H0: 1 = 2 = 3 vs. H1: At least two means differ Conclusion: Reject the null hypothesis. Can say that the average GMAT score differs between the three schools, according to this data. PTS: 1 REF: SECTION 14.1 69. In a completely randomized design, 15 experimental units were assigned to each of four treatments. Fill in the blanks (identified by asterisks) in the partial ANOVA table shown below. Source of Variation Treatments Error Total SS * * 2512 df * * * MS 240 * F * ANS: Source of Variation Treatments Error Total SS 720 1792 2512 df 3 56 59 MS 240 32 F 7.5 PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 70. In a completely randomized design, 12 experimental units were assigned to the first treatment, 15 units to the second treatment, and 18 units to the third treatment. A partial ANOVA table is shown below: Source of Variation Treatments Error Total a. b. SS * * * MS * 35 F 9 Fill in the blanks (identified by asterisks) in the above ANOVA table. Test at the 5% significance level to determine if differences exist among the three treatment means. ANS: a. Source of Variation Treatments Error Total b. df * * * SS 630 1470 2100 df 2 42 44 MS 315 35 F 9 H0: 1 = 2 = 3 vs. H1: At least two means differ Rejection region: F > F0.05,2,42 3.23 Test statistics: F = 9.0 Conclusion: Reject the null hypothesis. Yes, differences in means exist in at least two of the three treatment means, according to this data. PTS: 1 REF: SECTION 14.1 Mutual Funds An investor studied the percentage rates of return of three different types of mutual funds. Random samples of percentage rates of return for four periods were taken from each fund. The results appear in the table below: Mutual Funds Percentage Rates Fund 1 Fund 2 Fund 3 12 4 9 15 8 3 13 6 5 14 5 7 17 4 4 71. {Mutual Funds Narrative} Set up the ANOVA Table. Use = 0.05 to determine the critical value. ANS: Source of Variation Treatments Error Total PTS: 1 SS 252.40 49.20 301.60 df 2 12 14 MS 126.20 4.10 F 30.78 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 72. {Mutual Funds Narrative} Test at the 5% significance level to determine whether the mean percentage rates for the three funds differ. ANS: H0: 1 = 2 = 3 vs. H1: At least two means differ Rejection region: F > F0.05,2,12 = 3.89 Conclusion: Reject the null hypothesis. Yes, the mean percentage rates differ for at least two of the three mutual funds, according to this data. PTS: 1 REF: SECTION 14.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher.